TRANCHING, CDS AND ASSET PRICES: HOW FINANCIAL...

TRANCHING, CDS AND ASSET PRICES: HOW FINANCIAL INNOVATION CAN CAUSE BUBBLES AND CRASHES

By

Ana Fostel and John Geanakoplos

July 2011 Revised August 2011

COWLES FOUNDATION DISCUSSION PAPER NO. 1809R

COWLES FOUNDATION FOR RESEARCH IN ECONOMICS YALE UNIVERSITY

Box 208281 New Haven, Connecticut 06520-8281

http://cowles.econ.yale.edu/

Tranching, CDS and Asset Prices: How Financial

Innovation can Cause Bubbles and Crashes.

Ana Fostel John Geanakoplos ∗

August, 2011

Abstract

We show how the timing of financial innovation might have contributedto the mortgage bubble and then to the crash of 2007-2009. We show whytranching and leverage first raised asset prices and why CDS lowered themafterwards. This may seem puzzling, since it implies that creating a derivativetranche in the securitization whose payoffs are identical to the CDS will raisethe underlying asset price while the CDS outside the securitization lowers it.The resolution of the puzzle is that the CDS lowers the value of the underlyingasset since it is equivalent to tranching cash.

Keywords: Financial Innovation, Endogenous Leverage, Collateral Equi-librium, CDS, Tranching and Asset Prices.

JEL Codes: D52, D53, E44, G01, G10, G12

In this paper we propose the possibility that the mortgage boom and bust crisis

of 2007-2009 might have been caused by financial innovation. We suggest that the

astounding rise in subprime and Alt A leverage from 2000 to 2006, together with the

remarkable growth in securitization and tranching throughout the 1990s and early

2000s, raised the prices of the underlying assets like houses and mortgage bonds. We

further raise the possibility that the introduction of Credit Default Swaps, CDS, in

2005 and 2006 brought those prices crashing down with just the tiniest spark.

∗Fostel: George Washington University, 2115 G Street, Suite 370, Washington, DC, 20009,USA. [email protected]. Geanakoplos: Yale University Box 208281, New Haven, CT 06520-8281, USA and Santa Fe Institute, 1399 Hyde Park Road Santa Fe, New Mexico 8750, [email protected] thank audiences at AEA meetings and IMF Research Departmentfor useful comments. We also want to thank two anonymous referees for very useful comments.

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Securitization and tranching did not happen over night. The securitization of

mortgages by the government agencies Fannie Mae and Freddie Mac began in earnest

in the 1970s, when the first pools of mortgages were assembled and shares were sold

to investors. In 1986 Salomon and First Boston created the first tranches, buying

Fannie and Freddie Pools and cutting them into four pieces. This was no simple

task, because it involved not only special tax treatment by the government, but also

the creation of special legal entities and trusts which would collect the homeowner

payments and then divide them up among the bondholders. By the middle 1990s

the greatest mortgage powerhouse was the investment bank Kidder Peabody, cutting

hundreds of billions of dollars worth of mortgage pools into over 90 types of tranches

called CMOs (collateralized mortgage obligations). These tranches bore esoteric

names like floater, inverse floater, IO, PO, inverse IO, Pac, Tac, etc. The young

traders, often in their mid 20s, who collectively engineered this multi-trillion dollar

operation were motivated by the profits they could make buying pools of mortgages

and cutting them up into more valuable tranches. They would find out the needs

of various buyers and tailor make the tranches to deliver money in just those states

of nature that the buyers wanted them. In short, they exploited the heterogeneous

needs of their buyers by creating heterogeneous pieces out of a homogeneous pie.

The impetus driving the tranching machine was not a demand for riskless assets;

on the contrary, it was a demand for contingent assets. The Fannie and Freddie

principal mortgage payments were guaranteed against homeowner default by Fannie

and Freddie, enabling the tranches to be rated AAA. But that hardly meant they

were riskless. Changes in interest rates or changes in prepayments by the underlying

homeowners could radically alter the cash flows of the tranches. Gradually Wall

Street came to see that default risk was just one among many risks, and pools and

tranching began to be undertaken without government guarantees, for example for

jumbo mortgages that were not eligible for purchase by Fannie and Freddie and for

credit cards and other assets.

Spurred on by these private securitizations, Wall Street dreamt up the idea in

the mid 1990s of pooling and tranching subprime mortgages, with no government

guarantees at all. Through a cleverly constructed architecture of pooling, senior

pieces were still able to get AAA bond ratings because they were protected by junior

tranches that absorbed the losses in case of homeowner defaults. The subprime

mortgage market grew from a few million dollars to a trillion by 2006.

In the 1990s credit default swaps were invented for corporate bonds and sovereign

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bonds. It was not until 2005, however, that credit default swaps were standardized

for mortgages. A CDS is a kind of insurance on an asset or bond. It is the promise

to take back the underlying asset at par once there is a default, that is, to make up

the losses of the underlying asset.

Our approach, like many papers in economics that take technological innovation

as exogenous, is to take the financial innovations in the mortgage market between

1986 and 2010 as exogenous and investigate their consequences for asset pricing.

Under this view, the tranching of subprime mortgages couldn’t have begun earlier,

because it had to wait for the innovation of CMO tranching. In later work we hope

to explain why the innovations came when they did and why for example CDS seem

to appear in various markets only after the risk of default is generally recognized to

be significant.

The size of these financial innovations is certainly staggering, and leaves one

wondering what their effects might have been. Consider first the history of subprime

and Alt A leverage and housing prices from 2000-2008 shown in Figure 1, taken

from Geanakoplos (2010b). Leverage went from about 7 in 2000 (14 percent average

downpayment for the top half of households) to about 35 in the second quarter of

2006 (2.7 percent downpayment on average for the top half of households). Next

consider the growth of securitization and tranching presented in Figure 2, especially

in the late 1990s and early 2000s. These amounted to trillions of dollars a year.

Finally, consider the growth of the CDS market presented in Figure 3, especially

after 2005. The available numbers are not specific to mortgages, since most of these

were over the counter, but the fact that subprime CDS were not standardized until

late 2005 suggests that the growth of mortgage CDS in 2006 is likely even sharper

than Figure 3 suggests. What is clear is that the explosive growth of the CDS market

came after the explosive growth of securitization.

Many people who are aware of these numbers have linked securitization and CDS

to the crisis of 2007-2009. While we agree with much of their view, our analysis is

based on entirely different considerations. And we wish to explain the boom as well

as the bust.

Some problems that many critics have noted with tranching are i) the standing

opportunity for the original lender to sell his loans into a securitization destroys

his incentive to choose good loans and ii) once a pool is tranched it becomes very

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Figure 1: Housing Leverage.

difficult for the bond holders to negotiate with each other (for example to write down

principal).

Many observers have pointed to the creation of CDS as the source of many prob-

lems, to mention a few: i) important financial institutions wrote trillions of dollars of

CDS insurance; the economy could not run smoothly after they lost so much money

on their bad bets, ii) writers of CDS insurance did not even post enough collateral to

cover their bets, forcing the government to bail out the beneficiaries, iii) CDS were

traded on OTC markets, with a lack of transparency that enabled price gouging, iv)

CDS give investors (at least those who wrote much more insurance than the under-

lying assets were worth) the incentive to manipulate markets, for example to avoid

paying off a big insurance amount by directly paying off the bonds. George Soros

went as far as calling CDS “instruments of destruction that should be outlawed”

and claimed that “...some derivatives ought not to be allowed to be traded at all. I

have in mind credit default swaps. The more I’ve heard about them, the more I’ve

realized they’re truly toxic,”1

1Wall Street Journal, March 24, 2009.

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Figure 2: Securitization/Tranching.

The first main contribution of this paper is to show that tranching and leverage

raise the price of the underlying collateral even if they have no effect on the total cash

flows coming from the collateral. All the difficulties with tranching and CDS pointed

out by others in the last paragraphs rely on the pernicious effect of securitization

and tranching on the basic cash flows. So our thesis is quite different. But it is

really common sense. Indeed the historically enthusiastic government support for

the tranching of mortgages was doubtless due to an understanding that it raises the

price of the underlying mortgage assets and therefore reduces the borrowing costs to

the homeowner.

Tranching makes the underlying collateral more valuable because it can be broken

into pieces that are tailor made for different parts of the population, just as the

traders in the 1990s realized. Splitting plain vanilla into strawberry for one group

and chocolate for another should raise the value of the scarce ice cream. Leverage

is an imperfect form of tranching and one would guess that therefore leverage would

not raise the asset price as much as tranching.

We first build a static two-period model with heterogeneous agents in which

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Figure 3: CDS Markets. Outstanding notional amount.

we can investigate the circumstances under which these common sense conclusions

hold true. We compute equilibrium prices without leverage, with leverage, then with

tranching, then with tranching and CDS. We find that once the tranching technology

is invented and freely available, it will inevitably proceed in equilibrium all the way

to Arrow securities, or at least all the way to commonly verifiable events. Leverage

and asset tranching always raise asset prices above their CDS and no-leverage levels.

Somewhat surprisingly, however, we find that this fine tranching does not always

raise the asset price above the leverage price when all the general equilibrium effects

are taken into account. We prove that if there is more heterogeneity among the

pessimists than among the optimists, then tranching always yields a higher price

than leverage which in turn is higher than the no-leverage price.

Furthermore, we show that with tranching, the price of the underlying collateral

can rise above what any agent thinks it is worth, while with leverage the price of the

asset typically rises to what a more optimistic agent thinks it is worth. This tranched

hyper price fits the definition of a bubble given in Harrison and Kreps (1978), where

a bubble is defined as an asset price that is higher than any agent thinks the asset is

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worth. In Harrison and Kreps the explanation for bubbles turned on the ability of

agents to resell the asset to others who would think it was worth more, whereas in

our two-period model there is no resale of the asset.

Third, we show that the introduction of CDS dramatically lowers the asset price,

even below the non-leverage level, provided that the median investor thinks that the

asset is more than 50 percent likely to have a good payoff. This seems counterintuitive

at first glance. Tranching creates derivatives of the underlying asset, and presumably

one of those tranches could have exactly the same payouts as the CDS. Indeed that

happens in our model. The tranche and the CDS are perfect substitutes in every

agent’s mind. Yet when the CDS is created exclusively inside the securitization as a

tranche of the asset, it raises the asset price. When the CDS is created outside the

securitization it lowers the asset price. How could this be?

We show that on second thought this is not surprising at all. When agents sell

CDS and put up cash as collateral, they are effectively tranching cash! That raises

the value of cash relative to the reference asset. When every asset (more precisely,

when all future cash flows) can be perfectly tranched, we get the Arrow-Debreu

equilibrium, and all asset bubbles disappear. The depressing effect of CDS on asset

prices is most dramatic when the asset is not tranched, but is held outright or levered,

because in that case the buyers of the asset will divert their wealth into writing CDS,

which is a perfect substitute for holding the asset.

In Section I we present our two period model of collateral equilibrium. In Section

II we show how specifying the collateral technology in different ways allows the same

model to encompass leverage, tranching, and CDS in one simple framework. In

Section III we explain why leverage and securitization raise asset prices, and why

CDS lowers them. Finally in Section IV we describe a dynamic model in which a

non-levered initial situation is followed by the unexpected introduction of leverage,

then securitization and tranching and finally CDSs. All the way through very small

bad shocks are occurring. Nevertheless, prices rise dramatically during the initial

three phases, then come crashing down with the introduction of CDSs.

The timing of the financial innovation was crucial. Tranching and securitization

came first, raising asset prices, then CDS followed much later, crushing their prices.

Figures 1, 2 and 3 show empirical evidence suggesting that this was indeed the timing

in securitization and CDS markets.2 The timing of innovation was disastrous because

2Academic papers describing the financial crisis all agree on this fact, as in Brunnermeier (2009),Gorton (2010), Gorton and Metrick (2010), Geanakoplos (2010) and Stultz (2009).

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it caused a crash, forcing many people into bankruptcy or underwater. Had the CDS

come at the same time as the securitization, asset prices would never have gotten so

high, and the crash would have been milder, as we show in Section IV.

Our financial innovation theory of booms and busts complements the Leverage

Cycle theory proposed in Geanakoplos (2003, 2010a and 2010b) and developed fur-

ther in Fostel-Geanakoplos (2008, 2010 and 2011). The financial innovation theory

is similar in many ways to the leverage cycle, but unlike that theory, it does not rely

on any kind of shock, much less on a shock that also increases the volatility of future

shocks. In the financial innovation story of the boom and bust told here, innovation

could have generated the entire cycle on its own, with no external triggers.

In the leverage cycle story of the recent boom and bust told in Geanakoplos (2010a

and 2010b), a prolonged period of low volatility led to high leverage and therefore

high asset prices and the concentration of assets in the hands of the most optimistic

investors. When bad news came in 2007 in the form of a spike in delinquencies

of subprime homeowners, it not only directly reduced prices but it also reduced

leverage, because it created more uncertainty about what would happen next, which

had an indirect effect on asset prices. The combination of bad news, losses by the

hyper-leveraged optimists, and plummeting leverage led to a huge fall in asset prices,

much bigger than could be explained by the bad news alone.

Geanakoplos (2010b) attributed the rise in leverage to many factors besides low

volatility. One of these was the almost explicit government guarantees to Government

Agencies like Fannie Mae and Freddie Mac, and another was the implicit guarantees

to the big banks who were too big to fail. Yet another was low interest rates and

the resulting pursuit of yield. But most important he said was securitization. Yet he

provided no model for the connection between securitization and leverage. Geanako-

plos 2010a, 2010b, also suggested that the introduction of standardized credit default

swaps in 2005 had a sharp negative impact on the prices of assets.

Here we make rigorous the connection between leverage, tranching, and asset

prices by extending the model in Geanakoplos (2003). In the language of Fostel-

Geanakoplos (2008), tranching increases the collateral value of the underlying asset.

Leverage is an imperfect form of tranching and so raises the underlying asset value

less than ideal tranching. CDS is a form of tranching cash, and so raises the relative

value of cash, thus lowering the value of the reference asset.

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Our paper is more generally related to a literature on leverage as in Araujo, Kubler

and Schommer (2011), Acharya and Viswanathan (2011), Adrian and Shin (2010),

Brunnermeier and Pedersen (2009), Cao (2010), Fostel and Geanakoplos (2008, 2010

and 2011), Geanakoplos (1997, 2003 and 2010), Gromb and Vayanos (2002) and

Simsek (2010). It is also related to work that studies the asset price implications

of leverage as Hindy (1994), Hindy and Huang (1995) and Garleanu and Pedersen

(2011).

Our paper is also part of a growing theoretical literature on CDS. Bolton and

Oehmke (2011) study the effect of CDS on the debtor-creditor relationship. The

proposition that CDS tends to lower asset prices was demonstrated in Geanakoplos

(2010a), and confirmed in exactly the same model by Che and Sethi (2010).

I. General Equilibrium Model with Collateral

The model is a two-period general equilibrium model, with time t = 0, 1. Uncertainty

is represented by a tree S = {0, U, D} with a root s = 0 at time 0 and two states of

nature s = U,D at time 1.

There are two assets in the economy which produce dividends of the consumption

good at time 1. The riskless asset X produces XU = XD = 1 unit of the consumption

good in each state, and the risky asset Y produces YU = 1 unit in state U and

0 < YD = R < 1 unit of the consumption good in state D. Figure 4 shows asset

payoffs.

Each investor in the continuum h ∈ H = (0, 1) is risk neutral and characterized

by a linear utility for consumption of the single consumption good x at time 1, and

subjective probabilities, (qhU , qh

D = 1−qhU). The von-Neumann-Morgenstern expected

utility to agent h is

Uh(xU , xD) = qhUxU + qh

DxD (1)

We shall suppose that qhU is strictly monotonically increasing and continuous in

h. Examples are qhU = 1 − (1 − h)2, qh

D = (1 − h)2 and qhU = h, qh

D = 1 − h.

Each investor h ∈ (0, 1) has an endowment of one unit of each asset at time 0 and

nothing else. Since only the output of Y depends on the state and 1 > R, higher

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Figure 4: Asset Payoffs.

h denotes more optimism. Heterogeneity among the agents stems entirely from the

dependence of qhU on h.

The reader may be aghast by the simplicity of the model, and in particular by

heterogeneous priors, risk neutrality and the lack of endowment of the consumption

good in states 1 and 2. We hasten to assure such a reader that we are using the

simplest model we can to illustrate our point. None of the results depend on risk

neutrality or heterogeneous priors. By assuming common probabilities and strictly

concave utilities, and adding large endowments in state D vs state U for agents

with high h and low endowments in state D vs state U for agents with low h, we

could reproduce the distribution of marginal utilities we get from differences in prior

probabilities. We have chosen to replace the usual marginal analysis of consumers

who have interior consumption with a continuum of agents and a marginal buyer.

Our view is that the slightly unconventional modeling is a small price to pay for the

simple tractability of the analysis.

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A. Arrow Debreu Equilibrium

Arrow-Debreu equilibrium is easy to describe for our simple economy. It is given by

present value consumption prices (pU , pD), which without loss of generality we can

normalize to add up to 1, and by consumption (xhU , xh

D)h∈H satisfying

1.∫ 10 xh

Udh = 1 + 1

2.∫ 10 xh

Ddh = 1 + R

3. (xhU , xh

D) ∈ BhW (pU , pD) = {(xU , xD) ∈ R2

+ : pUxU + pDxD ≤ pU(1+1)+ pD(1+

R)}

4. (xU , xD) ∈ BhW (pU , pD) ⇒ Uh(x) ≤ Uh(xh),∀h

The interpretation of Arrow-Debreu equilibrium is that at time 0 agents trade

contingent commodities forward. An agent with high h for example might sell a

future claim for D consumption in exchange for U consumption. It is taken for

granted that h will deliver the goods in D if that state occurs.

We can easily compute Arrow-Debreu equilibrium. Because of linear utilities and

the continuity of utility in h and the connectedness of the set of agents H = (0, 1), at

state s = 0 there will be a marginal buyer, h1, who will be indifferent between buying

the Arrrow U and the Arrow D security. All agents h > h1 will sell everything and

buy only the Arrow U security. Agents h < h1 will sell everything and buy only the

Arrow D security. This regime is showed is Figure 5.

At s = 0 aggregate revenue from sales of the Arrow U security is given by pU ×2.

On the other hand, aggregate expenditure on it by the buyers h ∈ [h1, 1) is given by

(1 − h1)(2pU + (1 + R)(1 − pU)). Equating we have

(2pU + (1 + R)(1 − pU))(1 − h1) = 2pU (2)

The next equation states that the marginal buyer is indifferent between buying

the Arrow U and the Arrow D security.

qh1U /pU = (1 − qh1

U )/(1 − pU) (3)

Hence we have a system of two equations and two unknowns: the price of the

Arrow U security, pU , and the marginal buyer, h1. For the probabilities qhU = 1 −

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h=1

h=0

h1

Optimist buyers of Arrow U security

Pessimist buyers of Arrow D security

h1 Marginal buyer

Figure 5: Arrow-Debreu Equilibrium Regime.

(1 − h)2 and R = .2, we get h1 = .33 and pU = .55. The implicit prices of X and Y

are pX = pU1 + pD1 = 1 and pY = pU1 + pDR = .64.

B. Financial Contracts and Collateral

The heart of our analysis involves contracts and collateral. In Arrow Debreu equi-

librium the question of why agents repay their loans is ignored. We suppose from

now on that the only enforcement mechanism is collateral.

At time 0 agents can trade financial contracts. A financial contract (A, C) consists

of both a promise, A = (AU , AD), and an asset acting as collateral backing it,

C ∈ {X, Y }. The lender has the right to seize as much of the collateral as will make

him whole once the promise comes due, but no more: the contract therefore delivers

(min(AU , CU), min(AD, CD)) in the two states. The significance of the collateral is

that the borrower must own the collateral C at time 0 in order to make the promise

A.

We shall suppose every contract is collateralized either by one unit of X or by

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one unit of Y . The set of promises j backed by one unit of X is denoted by JX and

the set of contracts backed by one unit of Y is denoted by JY . In the next section

we will analyze different economies obtained by varying the set J = JX ∪ JY .

We shall denote the sale of promise j by ϕj > 0 and the purchase of the same

contract by ϕj < 0. The sale of a contract corresponds to borrowing the sale price,

and the purchase of a promise is tantamount to lending the price in return for the

promise. The sale of ϕj > 0 units of contract type j ∈ JX requires the ownership

of ϕj units of X, whereas the purchase of the same number of contracts does not

require any ownership of X.

C. Budget Set

Each contract j ∈ JC will trade for a price πj. An investor can borrow πj today by

selling contract j in exchange for a promise of Aj tomorrow, provided he owns C.

We can always normalize one price in each state s ∈ S = {0, U, D}, so we take

the price of X in state 0 and the price of consumption in each state U ,D to be one.

Thus X is both riskless and the numeraire, hence it is in some ways analogous to

a durable consumption good like gold, or to money, in our one commodity model.

Given asset and contract prices at time 0, (p, (πj)j∈J), each agent h ∈ H decides

his asset holdings x of X and y of Y and contract trades ϕj in state 0 in order to

maximize utility (1) subject to the budget set defined by

Bh(p, π) = {(x, y, ϕ, xU , xD) ∈ R+ × R+ × RJX × RJY × R+ × R+ :

(x − 1) + p(y − 1) ≤ ∑j∈J ϕjπ

j

∑j∈JX max(0, ϕj) ≤ x,

∑j∈JY max(0, ϕj) ≤ y

xU = x + y − ∑j∈JX ϕjmin(Aj

U , 1) − ∑j∈JY ϕjmin(Aj

U , 1)

xD = x + yR − ∑j∈JX ϕjmin(Aj

D, 1) − ∑j∈JY ϕjmin(Aj

D, R)}

At time 0 expenditures on the assets purchased (or sold) can be at most equal to

the money borrowed selling contracts using the assets as collateral. The assets put

up as collateral must indeed be owned. In the final states, consumption must equal

dividends of the assets held minus debt repayment.

Notice that there is no sign constraint on ϕj; a positive (negative) ϕj indicates

the agent is selling (buying) contracts or borrowing (lending) πj. Notice also that

we are assuming that short selling of assets is not possible, x, y ≥ 0.

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D. Collateral Equilibrium

We suppose that agents are uniformly distributed in (0, 1), that is they are described

by Lebesgue measure. A Collateral Equilibrium in this economy is a price of asset

Y, contract prices, asset purchases, contract trade and consumption decisions by all

the agents ((p, π), (xh, yh, ϕh, xhU , xh

D)h∈H) ∈ (R+ ×RJ+)× (R+ ×R+ ×RJX ×RJY ×

R+ × R+)H such that

1.∫ 10 xhdh = 1

2.∫ 10 yhdh = 1

3.∫ 10 ϕh

j dh = 0 ∀j ∈ J

4. (xh, yh, ϕh, xhU , xh

D) ∈ Bh(p, π),∀h

5. (x, y, ϕ, xU , xD) ∈ Bh(p, π) ⇒ Uh(x) ≤ Uh(xh),∀h

Markets for the consumption good in all states clear, assets and promises clear

in equilibrium at time 0, and agents optimize their utility in their budget sets. As

shown by Geanakoplos and Zame (1997), equilibrium in this model always exists

under the assumptions we made so far.

E. Tranching

One of the most important financial innovations has been the “tranching” of assets or

collateral. In tranched securitizations the collateral dividend payments are divided

among a number of bonds which are sold off to separate buyers. So far in our analysi

we have assumed that each collateral can back just one promise, so tranching seems

out of the picture. But in fact the collateral holder gets the residual payments after

the promise is paid, so effectively we have been tranching into two bonds all along.

And with two states of nature, we shall show that there is no reason to have more

pieces. So as long as there is no restriction on the nature of the promise, our collateral

equilibrium includes tranching.

In practice houses have been tranched into first and second mortgages, and some-

times third mortgages. These tranches have the property that they all move in the

same direction: good news for the house value is good news for all the tranches. But

14

when mortgages are tranched, the tranche values often move in opposite directions.

The more a floater pays, the less an inverse floater pays and so on. Even when

the tranches of subprime mortgages appear to have the form of debt for the higher

tranches, and equity for the lower tranches, the presence of various triggers which

move cash flows from one tranche to another can make the payoffs go in opposite

directions.

Thus in what follows we shall assume in our analysis that tranching has reached

a degree of perfection that permits Arrow security tranches to be created.3 The

tranching of mortgages in the CMO revolution of the 1990s moved far along in that

direction. And to the extent that the mortgage principal amount is nearly as high

as the house price, as often occurred in the 2000s, the mortgage already includes the

entire future value of the house. Thus we shall not distinguish between tranching the

asset or tranching a mortgage written on the asset. In short we shall assume that

the tranching is directly backed by the asset.

II. Leverage, Securitization, and CDS

In this section we study the effect of leverage and derivatives on equilibrium by

considering four different versions of the collateral economy introduced in the last

section, each defined by a different set of feasible contracts J . We describe each

variation and the system of equations that characterizes the equilibrium. In the next

section we compare equilibrium asset prices across all the economies.

A. No-Leverage Economy

We consider first the simplest possible scenario where no promises at all can be made,

J = ∅. Agents can only trade assets Y and X. They cannot borrow using the assets

as collateral.

Let us describe the system of equations that characterizes the equilibrium. Be-

cause of the strict monotonicity and continuity of qhU in h, and the linear utilities

and the connectedness of the set of agents H = (0, 1), at state s = 0 there will be a

3Of course, in reality Arrow securities cannot be created. But the reason has to do with thelack of verifiability and the cost of writing complex contingencies into a contract, both of which areignored in our analysis.

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unique marginal buyer, h1, who will be indifferent between buying or selling Y . In

equilibrium it turns out that all agents h > h1 will buy all they can afford of Y while

selling all their endowment of X. Agents h < h1 will sell all their endowment of Y .

This regime is shown in Figure 6.

h=1

h=0

h1

Optimist buyers of the asset

Pessimist sellers of the asset

h1 Marginal buyer

Figure 6: Non-Leverage Economy Equilibrium Regime.

At s = 0 aggregate revenue from sales of the asset is given by p × 1.4 On the

other hand, aggregate expenditure on the asset is given by (1 − h1)(1 + p), which is

total income from the endowment of one unit of X, plus revenue from the sale of one

unit of asset Y by buyers h ∈ [h1, 1). Equating supply and demand we have

p = (1 − h1)(1 + p) (4)

The next equation states that the price at s = 0 is equal to the marginal buyer’s

valuation of the asset’s future payoff.

4All asset endowments add to 1 and without loss of generality are put up for sale even by thosewho buy it.

16

p = qh1U 1 + (1 − qh1

U )R (5)

Hence we have a system of two equations and two unknowns: the price of the

asset, p, and the marginal buyer, h1. For the probabilities qhU = 1 − (1 − h)2 and

R = .2, we get h1 = .54 and p = .83.

B. Leverage Economy

Agents now are allowed to borrow money to buy more of the risky asset Y . We

let them issue non-contingent promises using the asset as collateral. In this case

J = JY , and each Aj = (j, j) for all j ∈ J = JY . The following result regarding

leverage holds.

Proposition 1: Suppose that in equilibrium the max min contract j∗ = mins=U,D{Ys} =

R is available to be traded, that is j∗ ∈ J = JY . Then j∗ is the only contract traded,

and the risk-less interest rate is equal to zero, this is, πj∗ = j∗ = R.

Proof: See Geanakoplos (2003) and Fostel-Geanakoplos (2010 and 2011).

Leverage is endogenously determined in equilibrium. In particular, the proposi-

tion derives the conclusion that although all contracts will be priced in equilibrium,

the only contract actively traded is the max min contract, which corresponds to

the Value at Risk equal zero rule, V aR = 0, assumed by many other papers in the

literature. Hence there is no default in equilibrium.

Taking the proposition as given, let us describe the system of equations that

characterizes the equilibrium. As before, there will be a marginal buyer, h1, who will

be indifferent between buying or selling Y . In equilibrium all agents h > h1 will buy

all they can afford of Y , i.e., they will sell all their endowment of the X and borrow

to the max min using Y as collateral. Agents h < h1 will sell all their endowment of

Y and lend to the more optimistic investors. The regime is showed in Figure 7.

At s = 0 aggregate revenue from sales of the asset is given by p × 1. On the

other hand, aggregate expenditure on the asset is given by (1− h1)(1 + p) + R. The

first term is total income (endowment of X plus revenues from asset sales) of buyers

h ∈ [h1, 1). The second term is borrowing, which from proposition 1 is R (recall that

the interest rate is zero). Equating we have

17

h=1

h=0

Optimist buyers/leveraged

Pessimist sellers/lenders

Marginal buyer h1 h1

Figure 7: Leverage Economy Equilibrium Regime.

p = (1 − h1)(1 + p) + R (6)

The next equation states that the price at s = 0 is equal to the marginal buyer’s

valuation of the asset’s future payoff.

p = qh1U 1 + (1 − qh1

U )R (7)

We have a system of two equations and two unknowns: the price of the asset,

p, and the marginal buyer, h1. Notice how equation (6) differs from equation (4).

Optimists now can borrow R. This will imply that in equilibrium a fewer number of

optimists can afford to buy all the asset in the economy. Hence, the marginal buyer

in the Leverage economy will be someone more optimistic than the marginal buyer

in the No-Leverage economy. We will discuss this in detail in Section III. For the

probabilities qhU = 1 − (1 − h)2 and R = .2, we get h1 = .63 and p = .89.

Finally, notice that buying the asset while leveraging to the max min is equivalent

to buying the Arrow U security. Since the owner needs to pay back R in period 1, his

18

net payoffs are 1−R at U and 0 at D. Hence, optimistic investors who are desperate

to transfer their wealth to the U state can very effectively do that by leveraging the

asset to the max min. In equilibrium, the implicit price of the Arrow U security is

given by pU = p−R1−R

= .86.

C. Asset-Tranching Economy

In this economy we suppose that the risky asset Y can be tranched into arbitrary

contingent promises, including the riskless promises from the last section and all

Arrow promises. The holder of the asset can sell off any of the tranches he does

not like and retain the rest. This is a step forward from the leverage economy, in

which investors holding a leveraged position on the asset could synthetically create

the Arrow U security. Now they can also synthetically create the Arrow D security.

To simplify the analysis we suppose at first that J = JY consists of the single

promise A = (0, R), tantamount to a multiple of the Arrow D security. Notice that

by buying the asset Y and selling off the tranche (0, R), any agent can obtain the

Arrow U security. Our parsimonious description of J = JY therefore already includes

the possibility of tranching Y into Arrow securities. We shall see shortly that once

that is possible, there is no reason to consider further tranches.

Let us describe the system of equations that characterizes the tranching equilib-

rium with the single tranche A = (0, R). In this case it is easy to see that there will

be two marginal buyers h1 and h2. In equilibrium all agents h > h1 will buy all of

Y , and sell the down tranche A = (0, R), hence effectively holding only the Arrow

U security. Agents h2 < h < h1 will sell all their endowment of Y and purchase all

of the durable consumption good X. Finally, agents h < h2 will sell their assets Y

and X and buy the down tranche from the most optimistic investors. The regime is

showed in Figure 8.

The system of equations that characterizes equilibrium is the following. Let πD

denote the price of the down tranche. Equation (8) states that money spent on the

asset should equal the aggregate revenue from its sale. The top 1 − h1 agents are

buying the asset and selling off the down tranche. They each have wealth 1 + p plus

the revenue from the tranche sale πD. Finally, there is 1 unit of total supply of the

asset. Hence we have

(1 − h1)(1 + p) + πD = p (8)

19

h=1

h=0

h1

Optimists: buy asset and sell Arrow Down tranche (hence holding the Arrow Up tranche)

Pessimists: buy the Arrow Down tranche.

h1


Moderates: hold the durable good.

Marginal buyer

Figure 8: Asset Tranching Economy Equilibrium Regime.

Notice that the implicit price of the Arrow U security, which the top 1−h1 agents

are effectively buying, equals pU = p − πD, the price of the asset minus the price of

the down tranche A = (0, R).

Equation (9) states that total money spent on the down tranche should equal

aggregate revenues from their sale. The bottom h2 agents spend all their endowments

to buy all the down tranches available in the economy (which is one since there is

one asset), at the price of πD.

h2(1 + p) = πD (9)

Equation (10) states that h1 is indifferent between buying the Arrow U security

and holding the durable consumption good. So his expected marginal utility from

buying the Arrow U security, the probability qh1U multiplied by the delivery of 1,

divided by its price, p − πD, equals the expected marginal utility of holding the

durable consumption good divided by its price, 1.

20

qh1U

p − πD

= 1 (10)

Finally, equation (11) states that h2 is indifferent between holding the down

tranche and the durable consumption good X. So his expected marginal utility from

buying the down tranche, which is the probability 1−qh2U multiplied by the payoff R,

divided by its price, πD, equals the expected marginal utility of holding the durable

consumption good divided by its price, 1.

(1 − qh2U )R

πD

= 1 (11)

We have a system of four equations and four unknowns: the price of the asset, p,

the price of the down tranche πD, and the two marginal buyers, h1 and h2.

Finally, notice that despite the fact that both Arrow securities are present, mar-

kets are not complete. Arrow securities are created through the asset. Hence, agents

cannot sell all the Arrow securities they desire and the Arrow-Debreu allocation

cannot be implemented. Tranching the asset is not enough to complete markets.

For the probabilities qhU = 1−(1−h)2 and R = .2, we get h1 = .58, h2 = .08, p = 1

and πD = .17. The asset price is much higher even than it was with leverage. The

simple reason is that leverage is an imperfect form of tranching. When the owner of

the asset Y can create pieces even better suited to heterogenous buyers it makes the

asset still more attractive.

One important conclusion to be drawn from combining equations (10) and (11)

is that the tranching asset price is

p = qh1U 1 + qh2

D R (12)

Interestingly, the asset price can be higher than any agent in the economy thinks

it is worth!. Defining the implicit Arrow security prices pU = p − πD = .83 and

pD = πD/R = .83, we see that p = pU + pDR > 1. We discuss how this could happen

in Section III.

A moment’s reflection should convince the reader that in our two state economy,

completely tranching Y is tantamount to allowing the asset to back a promise of R in

21

the down state. The asset holder on net then retains the U Arrow security. By buying

y units of Y and selling off y units of the tranche A = (0, R), and also buying z/R

units of the down tranche (perhaps created by somebody else), any agent who has

enough wealth can effectively purchase the arbitrary consumption xU = y, xD = z.

If it were possible to create different tranches beyond the two Arrow securities, no

agent would have anything to gain by doing so. In the end his new tranches would

not offer a potential buyer anything the buyer could not obtain for himself via the

Arrow tranches, as we just saw. With unlimited and costless tranching, tranching

into Arrow securities always drives out all alternative tranching schemes, a point

made in Geanakoplos-Zame (2011). Since Y = {(0, R)} already embodies Arrow

tranching, there is no reason to consider any more complicated tranching schemes.

D. CDS Economy

A CDS on the asset Y is a contract that promises to pay 0 at s = U when Y pays 1

and promises 1−R at s = D when Y pays only R. Figure 9 describes a comparison

between the underlying asset payoffs and the CDS payoffs.

��

��

��

��

��

��

��

��

Figure 9: CDS Payoffs.

22

A CDS is thus an insurance policy for Y. A seller of a CDS must post collateral,

typically in the form of money. In a two period model buyers of the insurance would

insist on 1−R of X as collateral. Thus for every one unit of payment, one unit of X

must be posted as collateral. We can therefore incorporate CDS into our economy

by taking JX to consist of one contract called c promising (0, 1).

We shall maintain our hypothesis that the asset Y itself can be tranched, so we

continue to suppose that JY consists of the single promise (0, R) called the down

tranche or D. Of course that is equivalent to supposing that (1 − R)/R units of the

asset Y can be put up as collateral for 1 CDS promising 1 − R in state D. In other

words, the down tranche in the securitization of Y is identical to the CDS. Yet we

shall show that the two have very different effects on the price of Y.

Equilibrium requires that buyers recognize that D and c, the CDS, are essentially

proportional and hence in equilibrium their prices must be in the same proportion.

Equation (13) states that

πD = Rπc (13)

Once equation (13) holds, it must also be the case that buyers recognize that

there are two equivalent ways of effectively buying the Arrow U security: tranching

the asset Y and tranching cash X. Hence we must have equation (14)

1

p − πD

=1

1 − πc

(14)

Given these identities, it is evident that in equilibrium there will be a marginal

buyer h1 such that all agents h > h1 will buy all of Y and X and sell the down

tranche D backed by the Y and the CDS contract c collateralized with X. Agents

h < h1 will sell all their endowment of Y and X and buy D and c. The regime is

showed in Figure 10.

Equation (15) states that the total money spent on Y and X has to equal the

revenues from their sale. Agents 1 − h1 buy both. They use their endowments as

before but now they also receive income from the sales of D and c, using Y and X

as collateral respectively. The total wealth represented by the endowments of X and

Y is 1 + p per person, and the total revenue from the sale of D and C is πD + πc.

This must equal the purchase cost of all the X and Y in the economy, which is also

1 + p. Hence

23

h=1

h=0

Optimists: buy Y and X and sell CDS and D tranche

Pessimists: buy CDS and D tranche


Figure 10: CDS Economy Equilibrium Regime.

(1 − h1)(1 + p) + πD + πc = 1 + p (15)

Equation (16) states that the marginal buyer should be indifferent between buying

the Arrow U security (either way he can) and buying the down tranche or the CDS.

qh1U

p − πD

=(1 − qh1

U )

πc

(16)

A succinct way of describing the difference between this economy and the previous

one is that CDS allow for the tranching of cash in addition to the previous tranching

of assets. As a consequence, with only two states of the world, CDS and tranching

allow the economy to implement the Arrow-Debreu equilibrium. Hence, for the

probabilities qhU = (1 − h)2 and R = .2, we get the same equilibrium as the one

described in the Arrow-Debreu section I.A. The price of the asset is given by p =

pU + RpD = .64.

Finally, a CDS can be “covered” or “naked” depending on whether the buyer of

24

the CDS needs to hold the underlying asset. Our previous discussion corresponds

to the case of “naked” CDS. When CDS must be “covered”, agents willing to buy

CDS need to hold the asset. But notice that holding the asset and buying a CDS is

equivalent to holding the risk-less bond, which was already available without CDS.

“Covered” CDS have no effect on equilibrium. For the rest of the paper we will focus

only on the “naked” CDS case.

III. Financial Innovation and Asset Pricing

We solve for equilibrium with probabilities qhU = 1 − (1 − h)2 in all the economies

just described as R varies. Figure 11 displays the Y asset prices p for different values

of R.5

� ��

� ��

� ��

� ��

� ��

� ��

� ��

� ��

� ��

��

��

��

��

� ��

��

Figure 11: Asset Prices in all economies for different values of R.

For all economies the asset price increases as R increases and disagreement dis-

appears. This is not surprising, since the asset clearly makes more payments the

5Complete results are presented in Table 1 in Appendix C.

25

higher is R and so naturally its price should increase. We come back to how far it

should increase shortly.

By far the most important implication of our numerical simulations is that lever-

age and tranching make the asset price higher than it would be without leverage,

and still higher than it would be in the CDS or Arrow-Debreu economy. Leverage

and derivatives thus have a profound effect on asset prices. And therefore so does

financial innovation. We now investigate how general these results are.

Proposition 2: The asset price in the Leverage economy is higher than in the No-

Leverage economy for all strictly monotonic and continuous qhU , and all 0 < R < 1.

Proof: From equation (4) we see that in the No-Leverage economy

1

hNL1

= 1 + pNL

while in the Leverage economy, from equation (6),

1

hL1

+R

hL1

= 1 + pL

Assuming R > 0, pNL ≥ pL only if hL1 > hNL

1 . But from equations (5) and (7) (which

say that the asset price is equal to the marginal buyer’s valuation) these last two

inequalities are not compatible.

As discussed before, the possibility of borrowing against the asset makes it possi-

ble for fewer investors to hold all the asset in the economy. Hence, the marginal buyer

is someone more optimistic than in the No-Leverage economy, raising the price of the

asset. This effect was first identified in Geanakoplos (1997, 2003). This connection

between leverage and asset prices is precisely the Leverage Cycle theory discussed in

Geanakoplos (2003, 2010a, 2010b) and Fostel-Geanakoplos (2008, 2010 and 2011).

This theory can rationalize the housing market behavior during the crisis. Leverage

on housing increased dramatically from 2000 to 2006 and housing prices increased

dramatically during the same time. Leverage on housing collapsed in 2007 and the

same happened to housing prices.

The boost in the price of Y from leverage is greatest for intermediate values of

R, a region that can be characterized as “normal times” and when disagreement

is not negligible. For too low values of R, agents can borrow very little against

26

the asset and hence the marginal buyer will be someone very close to the marginal

buyer when borrowing is not possible. On the other extreme, when R is very high,

though borrowing is very important, agents almost agree on the outcome of the asset,

pushing even the no-leverage price up near to 1. Next we turn to tranching.

Proposition 3: The asset price in the Tranching economy is higher than in

the No-Leverage economy for all strictly monotonic and continuous qhU , and for all

0 < R < 1.

Proof: From equation (8) in the Tranching economy, we see that

1

hT1

+πT

D

hT1

= 1 + pT

From equation (4) we see that in the No-Leverage economy

1

hNL1

= 1 + pNL

Then pNL ≥ pT only if hL1 > hNL

1 . But from equations (5) and (12) these last two

inequalities are not compatible.

In the numerical simulations, tranching raised the asset price even above the

Leverage economy price. But this need not always be the case. Consider for example

the beliefs given by

qhU = max{1 − (1 − h)2, 1 − (1 − .60)2}

The Leverage economy equilibrium calculated earlier is still the same, since the

marginal buyer was hL1 = .63. But in the Tranching economy, the marginal buyer is

hT1 < .60 and the price will therefore be .84(1)+.16(.2) = .872 < .89. In general, hT

1 <

hL1 . The tranching price becomes higher than the leverage price when q

hT1

U +qhT2

D >> 1,

because then the tranching price pT = qhT1

U 1+qhT2

D R can be very large. In the example

just given the probabilities are cooked up so that qhT1

U = qhT2

U .

All this suggests that the tranching price is higher than the leverage price when

there is more heterogeneity at the bottom, among the pessimists, than there is at

the top among the optimists. Indeed the following is true.

Proposition 4: If the probabilities qhU are concave in h, as well as strictly mono-

tonic and continuous, then the asset price in the Tranching economy is higher than

27

in the Leverage economy for all 0 < R < 1.

Proof: From equations (7) and (12) and the fact that hT1 > hT

2 , we have that

pL = qhL1

U 1 + qhL1

D R = R + qhL1

U (1 − R)

pT = qhT1

U 1 + qhT2

D R > R + qhT1

U (1 − R)

Assume temporarily that pL ≥ pT . Then we must have qhL1

U > qhT1

U , so hL1 > hT

1 ,

and hence (qhT1

D − qhL1

D ) > 0. Putting the above two equations together,

pT − pL = −(qhL1

U − qhT1

U )1 + [(qhT2

D − qhT1

D ) + (qhT1

D − qhL1

D )]R

To get our desired contradiction, it suffices to show that (qhL1

U − qhT1

U ) < (qhT2

D −q

hT1

D )R. From the hypothesized concavity of qh in h and from qhL1

D > qhT1

D > qhT2

D , it

suffices to show that hL1 − hT

1 < (hT1 − hT

2 )R. From equations (6) and (8) we know

that

(1 − hL1 )(1 + pL) + R = pL

1 + R

1 + pL= hL

1

(1 − hT1 )(1 + pT ) + πD = pT

1 + πD

1 + pT= hT

1

If pL ≥ pT , then

hL1 − hT

1 ≤ R − πD

1 + pT<

R

1 + pT

On the other hand, from equations (10) and (11) and Walras Law, we know that in

the tranching equilibrium, the agents between hT1 and hT

2 must hold all the X, hence

hT1 − hT

2 =1

1 + pT

From the last two equations and the concavity of qhU ,

qhL1

U − qhT1

U < R(qhT1

U − qhT2

U ) = R(qhT2

D − qhT1

D )

From the equation above and our earlier calculations, we conclude pT − pL > 0.

The idea of the proof is that the tranching price tends to be lower than the

28

leverage price because qhL1

U > qhT1

U , since hL1 > hT

1 . But the tranching asset price rises

because qhT2

D > qhT1

D since hT1 > hT

2 . By concavity the gap between hT1 and hT

2 has a

bigger effect than the gap between hL1 and hT

1 .

Our examples where qhU = 1 − (1 − h)2 or qh

U = h, both satisfy the concavity

hypothesis. A striking consequence of the power of securitization to raise the asset

price can be seen as R increases in our example with qhU = 1 − (1 − h)2. As the

graph shows, the price of the asset with tranching goes even above 1. This seems

puzzling since the durable good X delivers at least as much as the asset in every

state and its price in equilibrium equals 1. With leverage, the price rose because the

marginal buyer became a more optimistic agent, and the price came to reflect his

beliefs instead of the more pessimistic marginal buyer that obtained without leverage.

However, with leverage the asset price can never rise above 1, since no agent values

the asset at more than 1. But with tranching, the price can rise above what any agent

thinks it is worth. How can this be? The answer is that with tranching there are two

marginal buyers instead of one. The marginal buyer with leverage was indifferent

between the asset and the two cash flows into which leverage split it, and his beliefs

determined the price of the asset. In the tranching equilibrium the cash flows into

which the asset is split are held by different people, and nobody who wants one flow

would touch the other.

The tranching of the asset Y that was begun with leverage is perfected by tranch-

ing into Arrow securities. Leverage is a precursor or primitive form of tranching.

With leverage the asset can be used as collateral to issue non-contingent promises.

In the tranching economy, the asset can be used as collateral to issue contingent

promises raising even further its value as collateral. Y becomes so valuable because

it can broken into pieces that are tailor made for different parts of the population.

Splitting plain vanilla into strawberry for one group and chocolate for another raises

the value of the scarce ice cream.

This result parallels the result in Harrison and Kreps (78), who defined a bubble

as a situation in which an asset is priced higher than any agent thinks its cash

flows are worth. But we obtain the result in a static context without resale of the

asset. They displayed bubbles in a dynamic context with heterogeneous agents: the

most optimistic agent would buy the asset, but next period instead of suffering bad

cash flows he could resell the asset to a different agent. In our context, tranching

alone, without resale, creates collateral value and potentially bubbles. The gap in

29

prices between leveraged (or securitized) assets and unleveraged assets is what Fostel-

Geanakoplos (2008) called Collateral Value. When assets can be used as collateral to

borrow and not just as investment, there are deviations from classical forms of Law

of One Price. An asset with identical or inferior payoffs in the future can be priced

higher if it has higher collateral capacities.

This result is in tune with developments in the financial market. For more than

five years securitized mortgages traded at negative OAS.6 Securitization and tranch-

ing dramatically increased from the 1990s to 2006, along with leverage, explaining,

in our view, much of the rise in housing and related securities prices.

Finally, we turn our attention to the CDS Economy. The numerical simulations

show that CDS can dramatically lower the asset price, even below the no-leverage

level. In the simulations, the lower the R, the bigger is the price reduction from

the introduction of the CDS. For very high R, the introduction of CDS has little

effect on asset prices when compared to the no-leverage level. We have the following

proposition.

Proposition 5: The asset price in the CDS economy is lower than in the Lever-

age economy and lower than in the Tranching economy, for all strictly monotonic

and continuous qhU and for all 0 < R < 1.

Proof: Recall from equations (6), (8) that

1 + R

1 + pL= hL

1

1 + qhT1

D R

1 + pT= hT

1

Next, recall that the CDS equilibrium is the same as the Arrow-Debreu equi-

librium. Hence pCDS = pAD = pU1 + pDR. From equation (3), pD = qhCDS1

D . From

6Option Adjusted Spread (OAS) is the fudge factor Wall Street firms add to their pricing modelwhen all their risk factors are unable to explain market prices. Typically this number is positive,suggesting that the market is willing to pay less than the model values because of the fear of someunknown risk the modelers may have missed. But for mortgages around 2000 this OAS numberturned negative. We are suggesting here the reason is that FNMA and Freddie Mac were willingto pay more for the mortgages than their cash flows warranted because they could use them tocreate pools which would then be cut by Wall Street into more valuable tranches. An alternativeexplanation is that because of their implicit government guarantees, FNMA and Freddie Mac couldborrow at cheaper rates than anybody else and hence were willing to overpay for their assets.

30

equation (2) and Walras Law,

qhCDS1

D (1 + R)

1 + pCDS= hCDS

1

Suppose that pCDS ≥ pT . Then the marginal buyer hCDS1 > hT

1 , hence qhCDS1

D < qhT1

D .

But this contradicts the last two equations above. Similarly, if pCDS ≥ pL then the

marginal buyer hCDS1 > hL

1 , contradicting the first and third equations above.

The stunning fact that the introduction of CDS dramatically lowers the asset

price below the leverage price and below the tranching price seems counterintuitive

at first glance. Tranching creates exactly the same derivative payouts as the CDS.

They are perfect substitutes in every agent’s mind. Yet when the CDS is created

exclusively inside the securitization as a tranche of the asset, it raises the asset price.

When the CDS is created outside the securitization it lowers the asset price.

On second thought this is not surprising at all. CDS are a way of tranching X.

When agents sell CDS and put up cash as collateral, they are effectively tranching

cash! That raises the value of cash relative to other assets, lowering the others’

prices.

However, the CDS price need not be lower than the no-leverage price. Indeed

if agents think the bad state is very likely, then in the limit, by equation (4) and

Walras Law, we would find that hNL1 → 1/(1+R), while by equation (2) and Walras

Law, hCDS1 → 1. But if the median agent regards the good state as more likely than

the bad state, then the CDS price must be lower than the no-leverage price, as we

show in the next proposition.

Proposition 6: If the qhU are strictly monotonic and continuous in h and if

q1/2U ≥ 1/2, then the asset price in the CDS economy is lower than in the No-Leverage

economy for all 0 < R < 1.

Proof: Recall that in the No-Leverage economy, the top 1 − hNL1 agents hold

all the asset and the bottom hNL1 agents hold all the durable good X, which is more

valuable. Hence hNL1 > 1/2. In the CDS economy,

hCDS1 =

(1 + R)qhCDS1

D

2qhCDS1

U + (1 + R)qhCDS1

D

31

If hCDS1 ≥ 1/2, then by hypothesis q

hCDS1

U ≥ 1/2. But that contradicts the above

equation for R < 1. Hence hCDS1 < 1/2 < hNL

1 , and therefore pNL > pCDS.

Putting the last five propositions together, we get

Theorem: If the qhU are strictly monotonic, concave, and continuous in h and

if q1/2U ≥ 1/2, then the Tranching asset price is greater than the Leverage asset price

which is greater than the No-Leverage asset price which is greater than the CDS asset

price, for all 0 < R < 1.

In Appendix A we give the equations for two more economies, which allow us

to see the depressing effect of CDS on the asset price even more clearly. First we

consider what happens if cash X can be tranched by CDS, but Y cannot be tranched

or held as collateral for any kind of loan or derivative. Now the tranching boost to

price goes the other way, and the price of Y relative to X plummets still further.

Finally, we consider the situation which obtains today for many assets like sovereign

bonds. The underlying bond-asset is not tranched, but people can leverage their pur-

chases of it. On the other hand they can use cash as collateral to write CDS on the

asset-bond. As we see in Appendix A in Figure 16, the Y asset price is below the com-

plete markets, tranching-CDS price, but above the price where we have CDS backed

by cash alone. The reason is that the asset Y can only be imperfectly tranched via

leverage, whereas the cash is perfectly tranched.

IV. Dynamic Asset Prices: Bubbles and Crashes.

As we said at the outset, securitization emerged over a period of 20 years, and

then the CDS mortgage market suddenly exploded at the end of the securitization

boom. To model the implications of that dynamic we need to examine a multi-

period model. Needless to say, as long as nobody gets utility from holding collateral,

leverage, tranching, and CDS are irrelevant without uncertainty. Hence the dynamic

model must incorporate risk at each stage. To keep the model tractable (and thus to

avoid exponential growth in the number of states) we suppose that in every period

with high probability a tiny bit of bad news is received, which leaves the world a little

worse but much like it was, or with low probability all the uncertainty is resolved

and the good outcome obtains for sure from there on out. The most likely and most

interesting history occurs along the path of consecutive pieces of bad news. This

32

single history contains all the nodes at which the asset price is non-trivial. As we

shall see, this model is tractable, it assumes volatility increases with bad news (and

decreases with good news), and it makes leverage pro-cyclical.7

………….

1 1 1

t=0 t=1 t=N-1 t=N

��

��

��

Figure 12: Dynamic Asset Payoffs.

The risky asset payoffs are described in Figure 12, as are the probabilities of

the agents. At each point in time, there is either good news (with low probability

1 − (1 − h)2/N) or bad news (with high probability (1 − h)2/N). After good news,

uncertainty is completely resolved and the risky asset Y pays 1. However, after bad

news, the economy proceeds to the next period. After N consecutive periods of bad

news output materializes at R < 1. Notice that each agent believes that the final

output of Y will be R with probability (1− h)2 and 1 with probability 1− (1− h)2,

exactly as we had in the two period model of Section I. Each agent in the continuum

h ∈ (0, 1) begins at time 0 with one unit of Y and one unit of X, and has no

further endowments. As before, X produces 1 unit of the consumption good in every

7Fostel-Geanakoplos (2010) show that this stochastic structure is not something crazy or ad-hoc. They show that investors, given the opportunity to choose between technologies that exhibitpositive or negative correlations between first and second moments, they would mostly choose thelatter.

33

terminal state. We suppose that agents care only about expected consumption in

the terminal states.

Notice that as bad news comes, future output volatility increases. In other words,

bad news is revealed very slowly and each piece of bad news comes attached with a

spike in volatility. This type of technology, in which first moments and second mo-

ments of the asset distribution are negatively correlated, was introduced in Geanako-

plos (2003) in a three-period tree, and studied extensively by Fostel-Geanakoplos

(2010). This kind of economy induces pro-cyclical leverage, meaning that each piece

of bad news also decreases leverage. One potential drawback (or advantage) to the

model is that it presumes that if at the beginning one agent is more optimistic than

another, then he remains more optimistic throughout.8

The No-Leverage, Leverage, Tranching, and CDS economies are defined by their

contract structure, as in the static model. In each node s of the tree we are given

a set of one period contracts JX(s) using one unit of X as collateral and another

set of one period contracts JY (s) using one unit of Y as collateral. For the No-

Leverage economy, these sets are empty. For the Leverage economy, JX(s) is empty

and JY (s) consists of all promises (j, j) in the following two successor states. For

the Tranching economy JX(s) is empty and JY (s) consists of all possible promises

in the two successor states. In the CDS economy we would like to define the CDS as

a contract promising 1 − R in the terminal state following all bad news. But then

the question is: how much X collateral will the market require at each node s? The

answer is just enough to cover the market value of the CDS at node sD, the successor

node of s after bad news. With that insight we can reproduce the CDS equilibrium

by assuming that JX(s) consists of the one period promise (0, 1) for all s, and that

JY (s) consists of all possible promises in the two successor states.

Given the contract structure, equilibrium is defined by analogy to what we did

in Section I for the two period model in which N = 1. However, for longer hori-

zons agent optimization is much more complicated because agents must be forward

looking, anticipating what the price of Y will be in the successor states and taking

into account that they might not want to buy the risky asset Y today, even if its

expected payoffs exceed its price, because they might do better by waiting to buy it

next period when its price might be cheaper. All the equations that characterize the

equilibrium for each economy are presented in Appendix B.

8This may be the reason why in our model agents would buy CDS contracts at the beginning ifthey were available, rather than waiting until default becomes more likely.

34

Figure 13 shows the asset price evolution for the different economies for R = .2

and N = 10.9 Note first that at time t = 0 the leverage price and the tranching

price are much higher than they were at time t = 0 in the two-period economy.

Multiple periods increase the power of leverage and tranching to raise asset prices.

The no-leverage price at time t = 0 is exactly the same in the 10-period economy as

in the two-period economy.

�

��

� �

��

��

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��

��

��

��

�

��

��

� ��

��

Figure 13: Asset Price Dynamics for N = 10.

As before, the tranching price starts out higher than all the others, followed by

the leverage price, then the non-leverage price, then the very low CDS price. As time

goes and bad news keeps occurring, the prices all fall (there is no point in presenting

the price after good news, since that is always 1). Since all the price lines end up at

nearly the same price near .2, the fall is inversely related to the starting point. In the

Leverage economy prices start higher and hence the the price crash is much bigger

than in the No-Leverage and CDS economies. Since the no-leverage marginal buyer

never changes (he has no reason to sell after bad news), the price always reflects his

opinion of expected output. Thus, in the Leverage economy the bigger price decline

9Complete results are presented in Table 2 and 3 in Appendix C.

35

must be from feedback effects through deleveraging and through wealth loss among

the optimists. It is interesting that at the very beginning the leverage price is more

stable than the non-leverage price, but picks up steam fast as more bad news comes

in. However, the really dramatic fall in price on just a tiny bit of bad news happens

in the Tranching economy.

By contrast the fall is much more modest in the CDS economy. Early securitiza-

tion and tranching created a huge increase in asset prices. Had CDS been available

from the very beginning this over-valuation would not have happened.

However, as discussed in the introduction, CDS were actually introduced later.

Figure 14 shows the effect on asset prices if CDS were introduced to a Tranching

economy in period t = 2. We distinguish two different cases: a sudden, un-expected

introduction in which the economy is proceeding along as if CDS will never exist,

and an expected introduction in which from the very beginning it is known that CDS

will appear in period t = 2.

��

��

��

��

��

��

��

��

��

��

��

��

Figure 14: Late introduction of CDS at t = 2.

Without CDS, there would be a 17 percent drop in prices in the Tranching econ-

36

omy after two pieces of bad news, as we saw in Figure 13. This drop becomes much

bigger if CDS appear in period t = 2, namely 28 percent, even if the appearance of

CDS at time t = 2 is anticipated from the very beginning at time t = 0. The crash

becomes a horrific 46 percent if the CDS appear in period t = 2 as a surprise.

��

��

��$�

��%�

��&�

��

��

��$�

�'�� '�� '�� '(� �'$� �')� �'%� �'*� �'&� �'+�

��

�� '��

��'��

��'��

��'(�

Figure 15: Anatomy of the Bubble and Crash.

In our view the key to the size of the crisis is the order of the financial innova-

tion that materialized. Securitization, with all the tranching of CDOs and leverage

created a bubble and the introduction of CDS burst it, pushing pricing faster and

further down than they would have gone had there never been tranching or leverage

or CDS. Had CDS been there from the beginning, asset prices would never have

gotten so high.

The most dramatic drop in price occurs with the historical timing. In Figure 15

we see the evolution of the risky asset price assuming that the economy that starts

at t = 0 with no leverage. At t = 1 leverage is unexpectedly introduced and price

goes up, even though some bad news arrives. In t = 2 the risky asset tranching

technology is unexpectedly introduced and the price climbs even further, despite

more bad news. Finally, at t = 3 CDS are unexpectedly introduced and with it

37

the price crashes. Just one piece of bad news, on top of the introduction of CDS,

reduces the price by nearly 50 percent. In our dynamic model we only keep track

of price after bad news, yet the introduction of leverage and tranching is still strong

enough to offset the bad news and raise prices. Obviously in reality, the leverage

and tranching part of the cycle occurred mostly during pieces of “good news,” which

made the bubble even more violent.

It would be very interesting to endogenize the order of financial innovation. We

conjecture that with a transactions cost, CDS would not be traded at the beginning

because there is a very small probability of paying off. As the likelihood increases and

more disagreement is created, more people will want to trade them. This common

sense thought does not really obtain in our model, as far as we can see, because in

the model agents who are going to be the most pessimistic at the end are already

the most pessimistic at the beginning. They would buy their CDS from the start

if there were no transactions costs. Indeed CDS volume would decline, not increase

over time. Clearly this merits more investigation.

References

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Liquidity.” Journal of Finance, 66, 2011, 99-138.

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Financial Intermediation, 19 (3), 418-437, 2010.

• Araujo, Aloisio, Felix Kubler and Susan Schommer. 2011.“Regulating Collateral-

Requirements when Markets are Incomplete.” 2009. Forthcoming Journal of

Economic Theory.

• Brunnermeier, Markus. 2009. “Deciphering the Liquidity and Credit Crunch

2007-2009”. Journal of Economics Perspective, 23(1), 77-100.

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Empty Creditor Problem”. Forthcoming Review of Financial Studies.

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with Heterogenous Beliefs”. MIT job market paper.

38

• Che, Yeon and Rajiv Sethi R. 2010. “Credit Derivatives and the Cost of

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Volatility and Decrease Leverage”. Forthcoming Journal of Economic Theory.

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ing Complex System II, ed. W. Brian Arthur, Steven Durlauf, and David Lane,

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metric Society Monographs.

• Geanakoplos , John. 2010a. “The Leverage Cycle”, NBER Macro Annual, pp

1-65.

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Leverage Cycle”, Federal Reserve Bank of New York Economic Policy Review.

pp 101-131.

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Foundation Working Paper.

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kets.” Cowles Foundation Working Paper.

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agement (15)1 (January), 10-46.

39

• Gorton, Gary and Andrew Metrick. 2010. “Securitized Banking and the Run

on Repo.” Forthcoming Journal of Financial Economics.

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kets with financially consrained arbitrageurs” Journal of Financial Economics.

66,361-407.

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WP 2009-16.

A Two more CDS Economies

A.1 CDS without Tranching

For many assets, like sovereign bonds nowadays, direct tranching is not available but

CDS are written. As we said earlier, if the asset itself can be used as collateral by

the writer of CDS insurance, then indeed the assets are indirectly tranched, and the

analysis of the last section applies. There is good reason why the asset might be used

as collateral for its own insurance (puzzling as that sounds), because the optimists

most likely to own the bonds will be the same people writing the insurance. But

in practice cash seems to be the collateral of choice, so for completeness we analyze

also the case where only cash can be used as collateral for the CDS. We also do not

permit leverage of the assets. In the next section we do.

To formalize our assumption that there is no tranching and that CDS must be

collateralized with cash, we take J = JX to consist of just the CDS c that promises

40

(0, 1). In equilibrium there will be two marginal buyers h1 > h2. Agents h > h1 will

hold all of the “cash” X and use it all as collateral to write CDS c on Y , hence

effectively holding only the Arrow U security. Agents h2 < h < h1 will sell all their

endowment of X and purchase all of the asset Y . Finally, agents h < h2 will sell

their assets Y and X and buy the CDS from the most optimistic investors.

Equation (17) says that the top group of agents indeed holds all of X, and writes

the maximal number of CDS.

(1 − h1)(1 + p) + πc = 1 (17)

Note that these agents are effectively buying the Arrow Up security at an implicit

price of pU = 1 − πc. Equation (18) says that the bottom group of agents holds all

the CDS

h2(1 + p) = πc (18)

Equation (19) says that h1 is indifferent between writing CDS backed by X (and

thus synthetically holding the Arrow Up security) and holding the asset

qh1U

1 − πc

=qh1U + (1 − qh1

U )R

p(19)

Equation (20) says that h2 is indifferent between holding the asset and holding

the CDS c

qh2U + (1 − qh2

U )R

p=

1 − qh2U

πc

(20)

For the probabilities qhU = (1− h)2 and R = .2, we get h1 = .65, h2 = .29, p = .55

and πc = .36. The implicit arrow prices are given by pU = .63 and pD = .45.

A.2 CDS with Leverage

Finally we add the possibility of leverage for Y, maintaining the hypothesis that Y

cannot be tranched and that only X can be used as collateral for CDS. To formalize

this assumption, we take JX to consist of just the CDS c that promises (0, 1), and

we take JY to be all promises of the form (j, j) for any j > 0. As we saw before only

the promise (j∗, j∗) = (R, R) will be traded in equilibrium. We denote its price by

πj∗ .

41

In equilibrium there will as usual be two marginal buyers h1 > h2. Agents h > h1

will hold all of Y, purchased entirely via leverage, and all of the cash X, using it

all as collateral to write CDS c on Y , hence effectively holding only the Arrow U

security. Agents h2 < h < h1 will sell all their endowment of X and Y and lend it to

the optimists, protected by the collateral of Y . Finally, agents h < h2 will sell their

assets Y and X and buy the CDS from the most optimistic investors.

Equation (21) says that the top group of agents indeed holds all of Y via leverage

and also all of X, and writes the maximal number of CDS.

(1 − h1)(1 + p) + πc + πj∗ = 1 + p (21)

Note that these agents are effectively buying the Arrow Up security at an implicit

price of pU = 1−πc. But they are equally buying all the Arrow Up security via their

leveraged purchase of Y. Hence Equation (22) requires that

1 − R

p − πj∗=

1

1 − πc

(22)

Equation (23) says that the bottom group of agents holds all the CDS

h2(1 + p) = πc (23)

Equation (24) says that h1 is indifferent between writing CDS backed by X (and

thus synthetically holding the Arrow Up security) and lending

qh1U

1 − πc

=R

πj∗(24)

Equation (25) says that h2 is indifferent between lending and holding the CDS c

R

πj∗=

1 − qh2U

πc

(25)

For the probabilities qhU = (1 − h)2 and R = .2, we get h1 = .38, h2 = .28, p =

.61, πc = .45 and πj∗ = .17. The implicit arrow prices are given by by pU = .54 and

pD = .56.

Figure 16 presents asset prices values for different values of R as we did in the

main text but for the six different cases.

42

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��

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Figure 16: Asset Prices in all economies for different values of R.

B Equilibrium Equations for different Economies

in Section IV

1. CDS and Tranching Economy.

As noted before, this equilibrium corresponds to the Arrow-Debreu Equilib-

rium. The equations are the traditional equations in which we solve for arrow-

prices for N + 1 final states.

2. No-Leverage Economy.

We use the fact that the marginal buyer rolls over his debt at every node to

build up the system and then verify that the guess is correct. Notice that the

probability of good news in period k is given by (1 − (1 − hk)2)1/N .

p1 = ((1 − (1 − hk)2)1/N)N + (1 − ((1 − (1 − hk)

2)1/N)N)R

43

p1 =(1 − h1) + R

h1

...

pk = ((1 − (1 − hk)2)1/N)N−k + (1 − ((1 − (1 − hk)

2)1/N)N−k)R

...

pN−1 = ((1− (1− hN−1)2)1/N)N−(N−1) + (1− ((1− (1− hN−1)

2)1/N)N−(N−1))R

3. Leverage Economy.

Notice that since the final probability of disaster is constant (regardless of N),

the probability of bad news in period k is given by (1−hk)2/k. What is crucial

is that the marginal buyers are decreasing in time. There is a perpetual wealth

redistribution towards more pessimistic investors in each period after bad news

since previous leveraged investors go bankrupt.

pN+1 = R

pN = (1 − (1 − hN)2/N) + (1 − hN)2/NR

hN−1 =hN(1 + pN)

1 + pN+1

pN−1 =(1 − (1 − hN−1)

2/N) + (1 − hN−1)2/N (1−(1−hN−1)2/N )

(1−(1−hN )2/N )pN

(1 − (1 − hN−1)2/N) + (1 − hN−1)2/N (1−(1−hN−1)2/N )

(1−(1−hN )2/N )

hN−2 =hN−1(1 + pN−1)

1 + pN

...

p1 =(1 − (1 − h1)

2/N) + (1 − h1)2/N (1−(1−h1)2/N )

(1−(1−h2)2/N )p2

(1 − (1 − h1)2/N) + (1 − h1)2/N (1−(1−h1)2/N )

(1−(1−h2)2/N )

h0 =h1(1 + p1)

1 + p2

= 1

44

4. Tranching Economy

The probabilities are as before. The equilibrium in this economy is more chal-

lenging due to the multiplicity of marginal buyers in each period. It turns

out that both marginal buyers are decreasing with time. Buyers of Arrow U

security last only for one period. The second marginal buyer wealth evolution

turns out to be more complicated. They buy gold in the future, but eventually,

as time goes by, they buy Arrow U. In our example for N=10, h02 buys gold

from t = 1 to t = 8, and buys Arrow U in the last trading period. Define

qhU = (1 − (1 − h)(2/N)) and qh

D = 1 − qhU .

At t = 0

p0 − p1pD1 = (1 − h10)(1 + p0)

h20(1 + p0) = p1pD1

qh10

U

p0 − p1pD1

= qh10

U + qh10

D

qh10

U

p1 − p2pD2

qh20

D

pD1

C0 = qh20

U + qh20

D C0

where pD1 is the price of the Arrow down (which pays in period 1 after bad

news) and C0 is the continuation value of h20 given his future investment

For t = k for k = 2, ..., N − 2.

pk − pk+1pDk+1 =h1

k−1 − h1k

h1k−1 − h2

k−1

h2k

h2k−1

pk = pk+1pDk+1

qh1

kU

pk − pk+1pDk+1

= qh1

kU + q

h1k

D

qh1

kU

pk+1 − pk+2pDd+2

qh2

kD

pDk+1

Ck = qh2

kU + q

h2k

D Ck

At t = N − 1

45

pN−1 − pNpDN =h1

N−2 − h1N−1

h1N−2 − h2

N−2

h2N−1

h2N−2

pN−1 = pNpDN

qh1

N−1

U

pN−1 − RpDN

= 1

qh2

N−1

D

pDN

= 1

C Equilibrium Values in Section III and IV

Table 1, 2 and 3 show the complete equilibrium information from Section III and IV

respectively.

46

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